length of stroke performed with full pressure, which last call 1; the
hyperbolic logarithm of the quotient is the increase of efficiency due to
expansion. According to this rule it will be found, that if a given
quantity of steam, the power of which working at full pressure is
represented by 1, be admitted into a cylinder of such a size that its
ingress is concluded when one-half the stroke has been performed, its
efficacy will be raised by expansion to 1.69; if the admission of the steam
be stopped at one-third of the stroke, the efficacy will be 2.10; at
one- fourth, 2.39; at one-fifth, 2.61; at one-sixth, 2.79; at one-seventh,
2.95; at one-eighth, 3.08. The expansion, however, cannot be carried
beneficially so far as one-eighth, unless the pressure of the steam in the
boiler be very considerable, on account of the inconvenient size of
cylinder or speed of piston which would require to be adopted, the
friction of the engine, and the resistance of vapor in the condenser, which
all become relatively greater with a smaller urging force.

183. _Q._--Is this amount of benefit actually realized in practice?

_A._--Only in some cases. It appears to be indispensable to the realization
of any large amount of benefit by expansion, that the cylinder should be
enclosed in a steam jacket, or should in some other way be effectually
protected from refrigeration. In some engines not so protected, it has been
found experimentally that less benefit was obtained from the fuel by
working expansively than by working without expansion--the whole benefit
due to expansion being more than counteracted by the increased
refrigeration due to the larger surface of the cylinder required to develop
the power. In locomotive engines, with outside cylinders, this condition of
the advantageous use of expansion has been made very conspicuous, as has
also been the case in screw steamers with four cylinders, and in which the
refrigerating surface of the cylinders was consequently large.